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Mirrors > Home > MPE Home > Th. List > bcn1 | Structured version Visualization version GIF version |
Description: Binomial coefficient: 𝑁 choose 1. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
Ref | Expression |
---|---|
bcn1 | ⊢ (𝑁 ∈ ℕ0 → (𝑁C1) = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 11893 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | 1eluzge0 12286 | . . . . . . 7 ⊢ 1 ∈ (ℤ≥‘0) | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ (ℤ≥‘0)) |
4 | elnnuz 12276 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
5 | 4 | biimpi 218 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1)) |
6 | elfzuzb 12896 | . . . . . 6 ⊢ (1 ∈ (0...𝑁) ↔ (1 ∈ (ℤ≥‘0) ∧ 𝑁 ∈ (ℤ≥‘1))) | |
7 | 3, 5, 6 | sylanbrc 585 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 ∈ (0...𝑁)) |
8 | bcval2 13659 | . . . . 5 ⊢ (1 ∈ (0...𝑁) → (𝑁C1) = ((!‘𝑁) / ((!‘(𝑁 − 1)) · (!‘1)))) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁C1) = ((!‘𝑁) / ((!‘(𝑁 − 1)) · (!‘1)))) |
10 | facnn2 13636 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (!‘𝑁) = ((!‘(𝑁 − 1)) · 𝑁)) | |
11 | fac1 13631 | . . . . . . 7 ⊢ (!‘1) = 1 | |
12 | 11 | oveq2i 7161 | . . . . . 6 ⊢ ((!‘(𝑁 − 1)) · (!‘1)) = ((!‘(𝑁 − 1)) · 1) |
13 | nnm1nn0 11932 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
14 | 13 | faccld 13638 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 − 1)) ∈ ℕ) |
15 | 14 | nncnd 11648 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 − 1)) ∈ ℂ) |
16 | 15 | mulid1d 10652 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ((!‘(𝑁 − 1)) · 1) = (!‘(𝑁 − 1))) |
17 | 12, 16 | syl5eq 2868 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((!‘(𝑁 − 1)) · (!‘1)) = (!‘(𝑁 − 1))) |
18 | 10, 17 | oveq12d 7168 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((!‘𝑁) / ((!‘(𝑁 − 1)) · (!‘1))) = (((!‘(𝑁 − 1)) · 𝑁) / (!‘(𝑁 − 1)))) |
19 | nncn 11640 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
20 | 14 | nnne0d 11681 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (!‘(𝑁 − 1)) ≠ 0) |
21 | 19, 15, 20 | divcan3d 11415 | . . . 4 ⊢ (𝑁 ∈ ℕ → (((!‘(𝑁 − 1)) · 𝑁) / (!‘(𝑁 − 1))) = 𝑁) |
22 | 9, 18, 21 | 3eqtrd 2860 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁C1) = 𝑁) |
23 | 0nn0 11906 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
24 | 1z 12006 | . . . . 5 ⊢ 1 ∈ ℤ | |
25 | 0lt1 11156 | . . . . . 6 ⊢ 0 < 1 | |
26 | 25 | olci 862 | . . . . 5 ⊢ (1 < 0 ∨ 0 < 1) |
27 | bcval4 13661 | . . . . 5 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℤ ∧ (1 < 0 ∨ 0 < 1)) → (0C1) = 0) | |
28 | 23, 24, 26, 27 | mp3an 1457 | . . . 4 ⊢ (0C1) = 0 |
29 | oveq1 7157 | . . . . 5 ⊢ (𝑁 = 0 → (𝑁C1) = (0C1)) | |
30 | eqeq12 2835 | . . . . 5 ⊢ (((𝑁C1) = (0C1) ∧ 𝑁 = 0) → ((𝑁C1) = 𝑁 ↔ (0C1) = 0)) | |
31 | 29, 30 | mpancom 686 | . . . 4 ⊢ (𝑁 = 0 → ((𝑁C1) = 𝑁 ↔ (0C1) = 0)) |
32 | 28, 31 | mpbiri 260 | . . 3 ⊢ (𝑁 = 0 → (𝑁C1) = 𝑁) |
33 | 22, 32 | jaoi 853 | . 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑁C1) = 𝑁) |
34 | 1, 33 | sylbi 219 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁C1) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∨ wo 843 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 0cc0 10531 1c1 10532 · cmul 10536 < clt 10669 − cmin 10864 / cdiv 11291 ℕcn 11632 ℕ0cn0 11891 ℤcz 11975 ℤ≥cuz 12237 ...cfz 12886 !cfa 13627 Ccbc 13656 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-seq 13364 df-fac 13628 df-bc 13657 |
This theorem is referenced by: bcnp1n 13668 bcn2m1 13678 bcn2p1 13679 bcnm1 13681 bpoly2 15405 bpoly3 15406 bpoly4 15407 jm2.23 39586 |
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